The Millennium Prize Problems

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The Millennium Prize Problems

James Carlson,Arthur Jaffe,Andrew Wiles
The Millennium Prize Problems Book PDF

Author : James Carlson,Arthur Jaffe,Andrew Wiles
Publisher : American Mathematical Society, Clay Mathematics Institute
Page : 185 pages
File Size : 43,6 Mb
Release : 2023-09-14
Category : Mathematics
ISBN : 9781470474607

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The Millennium Prize Problems by James Carlson,Arthur Jaffe,Andrew Wiles Pdf

On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an influential role in mathematical research. A century later, on May 24, 2000, at a meeting at the Collège de France, the Clay Mathematics Institute (CMI) announced the creation of a US$7 million prize fund for the solution of seven important classic problems which have resisted solution. The prize fund is divided equally among the seven problems. There is no time limit for their solution. The Millennium Prize Problems were selected by the founding Scientific Advisory Board of CMI—Alain Connes, Arthur Jaffe, Andrew Wiles, and Edward Witten—after consulting with other leading mathematicians. Their aim was somewhat different than that of Hilbert: not to define new challenges, but to record some of the most difficult issues with which mathematicians were struggling at the turn of the second millennium; to recognize achievement in mathematics of historical dimension; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; and to emphasize the importance of working towards a solution of the deepest, most difficult problems. The present volume sets forth the official description of each of the seven problems and the rules governing the prizes. It also contains an essay by Jeremy Gray on the history of prize problems in mathematics.

The Millennium Problems

Keith J. Devlin
The Millennium Problems Book PDF

Author : Keith J. Devlin
Publisher : Granta Books
Page : 237 pages
File Size : 41,6 Mb
Release : 2005
Category : Mathematical recreations
ISBN : 1862077355

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The Millennium Problems by Keith J. Devlin Pdf

In 2000, the Clay Foundation of Cambridge, Massachusetts, announced a historic competition: Whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1million in prize money. They encompass many of the most fascinating areas of pure and applied mathematics, from topology and number theory to particle physics, cryptography, computing and even aircraft design. Keith Devlin describes here what the seven problems are, how they came about, and what they mean for mathematics and science. In the hands of Devlin, each Millennium Problem becomes a fascinating window onto the deepest questions in the field.

The Millennium Prize Problems

James A. Carlson,Arthur Jaffe,Andrew Wiles,Clay Mathematics Institute,American Mathematical Society
The Millennium Prize Problems Book PDF

Author : James A. Carlson,Arthur Jaffe,Andrew Wiles,Clay Mathematics Institute,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 192 pages
File Size : 52,9 Mb
Release : 2006
Category : Mathematics
ISBN : 082183679X

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The Millennium Prize Problems by James A. Carlson,Arthur Jaffe,Andrew Wiles,Clay Mathematics Institute,American Mathematical Society Pdf

"On May 24, 2000, at a meeting at the Collège de France, the Clay Mathematics Institute announced the creation of a US$7 million prize fund for the solution of seven important classic problems that have resisted solution. The prize fund is divided equally among the seven problems. There is no time limit for their solution. The Millennium Prize problems gives the official description of each of the seven problems and the rules governing the prizes"--Information screen.

The Poincaré Conjecture

Donal O'Shea
The Poincar Conjecture Book PDF

Author : Donal O'Shea
Publisher : Penguin UK
Page : 284 pages
File Size : 47,9 Mb
Release : 2008-10-30
Category : Science
ISBN : 9780141900346

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The Poincaré Conjecture by Donal O'Shea Pdf

The Poincaré Conjecture tells the story behind one of the world’s most confounding mathematical theories. Formulated in 1904 by Henri Poincaré, his Conjecture promised to describe the very shape of the universe, but remained unproved until a huge prize was offered for its solution in 2000. Six years later, an eccentric Russian mathematician had the answer. Here, Donal O’Shea explains the maths behind the Conjecture and its proof, and illuminates the curious personalities surrounding this perplexing conundrum, along the way taking in a grand sweep of scientific history from the ancient Greeks to Christopher Columbus. This is an enthralling tale of human endeavour, intellectual brilliance and the thrill of discovery.

Poincare's Prize

George G. Szpiro
Poincare s Prize Book PDF

Author : George G. Szpiro
Publisher : Penguin
Page : 324 pages
File Size : 48,6 Mb
Release : 2008-07-29
Category : Mathematics
ISBN : 9781440634284

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Poincare's Prize by George G. Szpiro Pdf

The amazing story of one of the greatest math problems of all time and the reclusive genius who solved it In the tradition of Fermat’s Enigma and Prime Obsession, George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri Poincaré developed the Poincaré Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldn’t prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found. Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.

Ricci Flow and the Poincare Conjecture

John W. Morgan,Gang Tian
Ricci Flow and the Poincare Conjecture Book PDF

Author : John W. Morgan,Gang Tian
Publisher : American Mathematical Soc.
Page : 586 pages
File Size : 52,6 Mb
Release : 2007
Category : Mathematics
ISBN : 0821843281

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Ricci Flow and the Poincare Conjecture by John W. Morgan,Gang Tian Pdf

For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. in 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjectu The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. in addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

The Great Mathematical Problems

Ian Stewart
The Great Mathematical Problems Book PDF

Author : Ian Stewart
Publisher : Profile Books
Page : 340 pages
File Size : 53,8 Mb
Release : 2013-03-07
Category : Mathematics
ISBN : 9781847653512

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The Great Mathematical Problems by Ian Stewart Pdf

There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics. The Great Mathematical Problems explains why these problems exist, why they matter, what drives mathematicians to incredible lengths to solve them and where they stand in the context of mathematics and science as a whole. It contains solved problems - like the Poincar Conjecture, cracked by the eccentric genius Grigori Perelman, who refused academic honours and a million-dollar prize for his work, and ones which, like the Riemann Hypothesis, remain baffling after centuries. Stewart is the guide to this mysterious and exciting world, showing how modern mathematicians constantly rise to the challenges set by their predecessors, as the great mathematical problems of the past succumb to the new techniques and ideas of the present.

What's Happening in the Mathematical Sciences

Barry Cipra
What s Happening in the Mathematical Sciences Book PDF

Author : Barry Cipra
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 51,8 Mb
Release : 2024-05-20
Category : Science
ISBN : 0821890433

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What's Happening in the Mathematical Sciences by Barry Cipra Pdf

Mathematicians like to point out that mathematics is universal. In spite of this, most people continue to view it as either mundane (balancing a checkbook) or mysterious (cryptography). This fifth volume of the What's Happening series contradicts that view by showing that mathematics is indeed found everywhere-in science, art, history, and our everyday lives. Here is some of what you'll find in this volume: Mathematics and Science Mathematical biology: Mathematics was key tocracking the genetic code. Now, new mathematics is needed to understand the three-dimensional structure of the proteins produced from that code. Celestial mechanics and cosmology: New methods have revealed a multitude of solutions to the three-body problem. And other new work may answer one of cosmology'smost fundamental questions: What is the size and shape of the universe? Mathematics and Everyday Life Traffic jams: New models are helping researchers understand where traffic jams come from-and maybe what to do about them! Small worlds: Researchers have found a short distance from theory to applications in the study of small world networks. Elegance in Mathematics Beyond Fermat's Last Theorem: Number theorists are reaching higher ground after Wiles' astounding 1994 proof: new developments inthe elegant world of elliptic curves and modular functions. The Millennium Prize Problems: The Clay Mathematics Institute has offered a million dollars for solutions to seven important and difficult unsolved problems. These are just some of the topics of current interest that are covered in thislatest volume of What's Happening in the Mathematical Sciences. The book has broad appeal for a wide spectrum of mathematicians and scientists, from high school students through advanced-level graduates and researchers.

Elliptic Tales

Avner Ash,Robert Gross
Elliptic Tales Book PDF

Author : Avner Ash,Robert Gross
Publisher : Princeton University Press
Page : 277 pages
File Size : 41,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780691151199

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Elliptic Tales by Avner Ash,Robert Gross Pdf

Describes the latest developments in number theory by looking at the Birch and Swinnerton-Dyer Conjecture.

Stalking the Riemann Hypothesis

Dan Rockmore
Stalking the Riemann Hypothesis Book PDF

Author : Dan Rockmore
Publisher : Vintage
Page : 306 pages
File Size : 52,5 Mb
Release : 2006-05-09
Category : Mathematics
ISBN : 9780375727726

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Stalking the Riemann Hypothesis by Dan Rockmore Pdf

For 150 years the Riemann hypothesis has been the holy grail of mathematics. Now, at a moment when mathematicians are finally moving in on a proof, Dartmouth professor Dan Rockmore tells the riveting history of the hunt for a solution.In 1859 German professor Bernhard Riemann postulated a law capable of describing with an amazing degree of accuracy the occurrence of the prime numbers. Rockmore takes us all the way from Euclid to the mysteries of quantum chaos to show how the Riemann hypothesis lies at the very heart of some of the most cutting-edge research going on today in physics and mathematics.

Mathematical Analysis of the Navier-Stokes Equations

Matthias Hieber,James C. Robinson,Yoshihiro Shibata
Mathematical Analysis of the Navier Stokes Equations Book PDF

Author : Matthias Hieber,James C. Robinson,Yoshihiro Shibata
Publisher : Springer Nature
Page : 471 pages
File Size : 52,5 Mb
Release : 2020-04-28
Category : Mathematics
ISBN : 9783030362263

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Mathematical Analysis of the Navier-Stokes Equations by Matthias Hieber,James C. Robinson,Yoshihiro Shibata Pdf

This book collects together a unique set of articles dedicated to several fundamental aspects of the Navier–Stokes equations. As is well known, understanding the mathematical properties of these equations, along with their physical interpretation, constitutes one of the most challenging questions of applied mathematics. Indeed, the Navier-Stokes equations feature among the Clay Mathematics Institute's seven Millennium Prize Problems (existence of global in time, regular solutions corresponding to initial data of unrestricted magnitude). The text comprises three extensive contributions covering the following topics: (1) Operator-Valued H∞-calculus, R-boundedness, Fourier multipliers and maximal Lp-regularity theory for a large, abstract class of quasi-linear evolution problems with applications to Navier–Stokes equations and other fluid model equations; (2) Classical existence, uniqueness and regularity theorems of solutions to the Navier–Stokes initial-value problem, along with space-time partial regularity and investigation of the smoothness of the Lagrangean flow map; and (3) A complete mathematical theory of R-boundedness and maximal regularity with applications to free boundary problems for the Navier–Stokes equations with and without surface tension. Offering a general mathematical framework that could be used to study fluid problems and, more generally, a wide class of abstract evolution equations, this volume is aimed at graduate students and researchers who want to become acquainted with fundamental problems related to the Navier–Stokes equations.

Unsolved Problems in Mathematical Systems and Control Theory

Vincent D. Blondel,Alexandre Megretski
Unsolved Problems in Mathematical Systems and Control Theory Book PDF

Author : Vincent D. Blondel,Alexandre Megretski
Publisher : Princeton University Press
Page : 352 pages
File Size : 55,5 Mb
Release : 2009-04-11
Category : Mathematics
ISBN : 9781400826155

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Unsolved Problems in Mathematical Systems and Control Theory by Vincent D. Blondel,Alexandre Megretski Pdf

This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science. The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed.

Quantum Field Theory and Manifold Invariants

Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann
Quantum Field Theory and Manifold Invariants Book PDF

Author : Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann
Publisher : American Mathematical Society, IAS/Park City Mathematics Institute
Page : 476 pages
File Size : 55,7 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9781470461232

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Quantum Field Theory and Manifold Invariants by Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann Pdf

This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.

The Golden Ticket

Lance Fortnow
The Golden Ticket Book PDF

Author : Lance Fortnow
Publisher : Princeton University Press
Page : 188 pages
File Size : 45,8 Mb
Release : 2017-02-28
Category : Computers
ISBN : 9780691175782

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The Golden Ticket by Lance Fortnow Pdf

The P-NP problem is the most important open problem in computer science, if not all of mathematics. Simply stated, it asks whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer. The Golden Ticket provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. Lance Fortnow traces the history and development of P-NP, giving examples from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. The Golden Ticket explores what we truly can and cannot achieve computationally, describing the benefits and unexpected challenges of this compelling problem.

Stable Stems

Daniel C. Isaksen
Stable Stems Book PDF

Author : Daniel C. Isaksen
Publisher : American Mathematical Soc.
Page : 159 pages
File Size : 45,9 Mb
Release : 2020-02-13
Category : Education
ISBN : 9781470437886

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Stable Stems by Daniel C. Isaksen Pdf

The author presents a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. He uses the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. He then uses the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. He also describes the complete calculation to the 65-stem, but defers the proofs in this range to forthcoming publications. In addition to finding all Adams differentials, the author also resolves all hidden extensions by 2, η, and ν through the 59-stem, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. The author also computes the motivic stable homotopy groups of the cofiber of the motivic element τ. This computation is essential for resolving hidden extensions in the Adams spectral sequence. He shows that the homotopy groups of the cofiber of τ are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows him to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.